Effects of Measurement Precision and Finite Numbers of Electrodes on Linear Impedance Imaging Algorithms

Impedance imaging systems reconstruct approximations to the conductivity inside a body from electrical measurements made on the boundary. The linearized problem was studied by Calderón, who showed that if the conductivity differs little from a constant, then it can be reconstructed approximately. In his work, Calderón assumed that arbitrary current densities can be applied anywhere on the body’s surface and the resulting voltages perfectly measured. In practice, however, imaging systems apply currents through a finite number of electrodes on the surface and measure the resulting voltages with a limited precision. In this paper, linear algorithms are described for approximating the electrical conductivity in a disk from finitely many limited precision measurements. The effects of finite precision and electrode number on a two-dimensional version of Calderón’s method are studied. It is shown that as the measurement errors go to zero and the electrodes proliferate in a space-filling manner, the finite electrode approximation converges to the approximation obtained by Calderón.